I have private health care through my employer, I've only used it once but it was very handy. I don't agree with your take at all, there's no way a private health care outfit could efficiently provide an effective emergency service across the entire UK for all possible injuries and illnesses, completely replacing or duplicating the NHS. The USA has tried this and it's an incredibly expensive disaster.

Employed middle aged workers in the USA pay about the same as we do for the NHS to support Medicare, Medicaid and CHIP through taxes, and then have to pay about the same again to actually get health care for themselves. Plus the system that costs twice as much as ours per citizen still leaves tens of millions of Americans without health insurance. It's utterly godawful. Our current system where everyone supports the NHS, and if you want top up services you pay for it, ensures solid funding for everyone's health care. The fact that my private care actually subsidises the NHS by reducing demand on it is a feature, not a bug. There's also the fact that private health care increases overall investment in health care, so in emergencies the NHS can call on private medical facilities which otherwise wouldn't be available. I know there's a fairness argument, but in practical terms our system benefits everyone.